Evaluation of dimension of fractal time series with the least square method

Properties of fractional Brownian motions （fBms） have been investigated by researchers in different fields, e.g. statistics, hydrology, biology, finance, and public transportation, which has helped us better understand many complex time series observed in nature [1-4]. The Hurst exponent H （0 〈 H 〈 1） is the most important parameter characterizing any given time series F（t）, where t represents the time steps, and the fractal dimension D is determined via the relation D = 2 - H.

Analytical Structure Matching and Very Precise Approach to the Coulombic Quantum Three—Body Problem

A powerful approach to solve the Coulombic quantum three-body problem is proposed. The approach is exponentially convergent and more efficient than the hyperspherical coordinate method and the correlation-function hyperspherical harmonic method. This approach is numerically competitive with the variational methods, such as that using the Hylleraas-type basis functions. Numerical comparisons are made to demonstrate the efficiency of this approach, by calculating the nonrelativistic and infinite-nuclear-mass limit of the ground state energy of the helium atom. The exponential convergency of this approach is due to the full matching between the analytical structure of the basis functions that are used in this paper and the true wavefunction. This full matching was not reached by most other methods. For example, the variational method using the Hylleraas-type basis does not reflects the logarithmic singularity of the true wavefunction at the origin as predicted by Bartlett and Fock. Two important approaches are proposed in this work to reach this full matching: the coordinate transformation method and the asymptotic series method. Besides these, this work makes use of the least square method to substitute complicated numerical integrations in solving the Schr?dinger equation without much loss of accuracy, which is routinely used by people to fit a theoretical curve with discrete experimental data, but here is used to simplify the computation.

A Computational Model for Velocity Separation in Shallow Sea

Based on the hydrodynamical feature and the theoretical velocity profiles of tidal flow and wind-induced flow in shallow sea, a computational model is established for the first time, which can separate observed velocity into tidal velocity and wind-induced velocity by use of the least square method. With the model, not only the surface. velocities of tidal. flow and wind-induced flow are obtained, but also the bed roughness height is determined and the wind velocity above the water surface is estimated. For verification of the model, the observed velocity in the Yellow River Estuary and the laboratory test is separated, then it is applied to the Yangtze River Estuary. All the results are satisfactory. The research results show that the model is simple in method, feasible in process and reasonable in result. The model is a valid approach to analysis and computation of field data, and can be applied to separate the observed velocity in shallow sea; at the same time, reasonable boundary conditions of the surface and bottom can be obtained for two- and three-dimensional numerical simulation.

A New Model for Velocity Profiles in Estuarine Waters

Water motion in estuarine waters is the result of the action of various dynamic factors. Firstly, based on the hydro- dynamic characteristics in estuarine waters, neglecting the nonlinear effects of various flow hydrodynamic factors, the logarithm velocity profile of tidal current and the cubic velocity profile of Hansen and Rattray （1965） made for linear super- position at a sense of first order, a new model for velocity profile in estuarine waters is established. Then, by introducing the least square method combination of enumeration, the velocity profile data of wind-driven current measured in the laboratory and that observed at the North and the South Branches of the Yangtze Estuary are verified and compared with other formulas, all with satisfactory results. The results show that the new model not only considers the influences of various dynamic factors, such as tide, wind force, run-off and density pressure with high accuracy, but also provides reasonable boundary conditions on the bottom for hydrodynamics numerical simulation in estuarine waters. Thereby, the accuracy and credibility of numerical computation and prediction of water flow are improved. The research is theoretically important for the estuarine hydrodynamics.